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1 A Study on the Motion Control of a Stabilizer System Using an Adaptiv e F uzzy Controller

2 A bs tract

3 / A ppendix 61

4 A Study on the Motion Control of a Stabilizer Sy stem U sing an A daptiv e Fuzzy Controller Tae-H oon K im Department of Control & Instrumentation Engineering, Graduate School, Korea Maritime University Abstract A tracking sy stem equipped on a fixed body needs the position al inform ation of the t arg et an d the control appar atu s to follow the azimuth angle and the elev ation angle of the moving object, when the tracking system is equipped on the moving v ehicle like a ship, it requires a st abilizin g sy stem t o flat the tracking system against the moving vehicle as w ell as the position al inform ation and the contr ol equipm ent. T he st abilizer sy stem compens ates the tr acking sy stem for the v ertical, horizont al an d direction al deviation s bet w een the tracking system and reference frame. T his st abilizer sy stem can be applied to a s atellite antenna on ships, a sun tracking sy stem on moving vehicles, and a camera

5 serv o control loop to t ake a st able im age ag ain st the vibr ation. In this paper, a stabilizer system using an active stabilization method is composed. An adaptive fuzzy controller is also sugg est ed, which is applicable t o sy st em s with structur al and parameter uncertainty. It is the 2nd/ 1st- type adaptive fuzzy control algorithm using adv antages of 1st - type and 2nd- type adaptiv e fuzzy alg orithm. Sev er al simulation s ar e ex ecut ed for v erifying the perform ance of the sugg est ed m ethod. T hrough experiments using a composed stabilizer system, tracking perform ances are ev aluated.

6 1, (Azimuth ) (Elevation ).,,. (Surge), (Sway), (Heave) (Roll), (Pitch ), (Yaw ) 6- (Six - degree of freedom movements).,., (Stabilizer system) [1-3]., T V,.,,,, [4,5]. (Active Stabilization method)

7 X, Y 2,. (Piezo- Electric Gyro Sensor ), DC [6]. (Adaptive fuzzy controller ) [7,8]. (Adaptive Law ) 1 (First - type Adaptive fuzzy algorithm ) (Fuzzy rule) 2 (Second- type Adaptive fuzzy algorithm ) 2/ 1 (Second/ fir st - type Adaptive fuzzy algorithm ) / 1, 4,

8 , X, Y, Z 3 X, Y, Z 3,, Figure 2.1 Component of six - degree of freedom movements - 3 -

9 6-.,,,.,.,. (Stabilization). (Stabilizer ),.. T V,.,,,. (Passive Stabilization Method) (Active Stabilization Method)

10 .., (Gimbal) 2, 2 DC. DC PWM (Pulse Width Modulation ) Figure 2.2 Structure of a two- axis stabilizer system - 5 -

11 X, Y 2,, 2 X, Y 2., X Y. 2.3 Figur e 2.3 Block diagr am of a st abilizer sy st em - 6 -

12 2.3 3.,,,. Murata. 2 X Y.. X, Y 2 2 DC ,,.. PWM DC.,, LCD [9,10]

13 2.1 Photo 2.1 Photograph of a stabilizer sy stem - 8 -

14 2.3. (Mechanical Gyros) (Optical Gyroscopes), (Coriolis Effect ) (Piezo- Electric Gyroscopes). (Drift Error ) Murata ENV - 05DB Gyrostar [deg/ sec], 2.5[V]. 2.4 Figur e 2.4 Output char acteristic curv e of a gyr o sen sor - 9 -

15 (Low Pass Filter ) A/ D.. X, Y Photo 2.2 Photogr aph of a gyr o sen sor mounted

16 2.4 2., X Y 2., DC. PWM.. A/ D. 2.5 Figur e 2.5 Structure of a m otion stabilizer

17 (a) Roll motion (b ) Pitch motion Photo 2.3 Phot ogr aph of a tw o- axis motion stabilizer

18 2.5 80C196KC,, LCD. A/ D..., DC PWM. A/ D [11-16]. 2.6 Figure 2.6 Block diagram of a data controller

19 Intel 16 80C196KC 8 A/ D DC PWM,. [17-19]. 20MHz 4 (HSO) 4 (HSI) 256 RAM 28 / s (16MHz) (Power down)/ (Idle Mode) 16 (W atchdog timer ) (Full duplex ) 8 / / / 8/ 10 A/ D PT S (Peripheral T ransaction Server ) 8 I/ O PWM

20 2.4 Photo 2.4 Phot ogr aph of a data contr oller

21 3 3.1 (Adaptive Fuzzy Controller ), [20]. (Fuzzy Logic System )., [7,8]. (Linguistic Information ) [2 1]..,.,. (Order ) (Bounds).. (Adaptive Law ).,. (Direct )

22 (Indirect )..,.. IF - T HEN. 1 (First- type) 2 (Second- type). (3.1) [22]. f (x ) = M l = 1 M l = 1 l [ [ n i = 1 F l i (x i) ] n i = 1 F l i (x i) ] (3.1), l, F l i (x i). l 1, l F l i (x i)

23 3.2, 3.1. (Fuzzy Basis Function) (Identification Model) Figure 3.1 Block diagram of the indirect adaptive fuzzy control sy stem

24 3.2.1 n x 1 = x 2, x 2 = x 3, x n = p(x 1,, x n ) + q(x 1,, x n ) u, y = x 1 (3.2) x ( n ) = p (x, x,, x ( n - 1) ) + q(x, x,,x ( n - 1) ) u, y = x (3.3), p(x), q(x ) :, u R, y R :, x = (x 1, x 2,, x n ) T = (x, x,,x ( n - 1) ) T R N. p (x ) q(x ) (3.4) (3.5). p ( x p ) = T p p ( x ) (3.4) T q( x q ) = q q ( x ) (3.5), = ( 1,, M ) T : l( x ) = M l = 1 [ n i = 1 F l i (x i) n i = 1 F l i (x i) ], ( x) = ( 1 ( x ),, M ( x )) T : [23]

25 3.2.2 y ( t) y m ( t). e = y m - y. h ( s) = s n + k 1 s ( n - k = ( k n,, k 1 ) T 1) + + k n R N. u = 1 q(x ) [ - p (x ) + y ( n ) m + k T e] (3.6) (3.6) (3.3) e = ( e, e,, e ( n - 1) ) T e ( n ) + k 1 e ( n - 1) + + k n e = 0 (3.7). lim t e( t) = 0. (3.6) p(x) q(x) p ( x p ) q( x q ) (Certainty Equivalent Control) u c. u c = 1 q( x q ) [ - p( x p ) + y ( n ) m + k T e] (3.8)

26 3.2.3 (3.3) (3.8). e ( n ) = - k T e + [ p( x p ) - p (x) ] + [ ( q( x q ) - q( x) ]u c (3.9) e = c e + b c [ ( p ( x p ) - p (x)) + ( q( x q ) - q(x )) u c ] (3.10), c = , b c = k n - k n k c Lyapunov (Positive Definite Symmetric Matrix ) P., Q [24]. T c P + P c = - Q (3.11) Lyapunov V e = 1 2 e T P e (3.12)

27 , (3.10) (3.12). V e = = e T P e e T P e (3.13) 1 2 e T Qe + e T P b c [ ( p ( x p ) - p(x )) + ( q( x q ) - q(x )) u c ] e 0 V e. V e 0 V e V. (3.13) V e 0 0. u c (Supervisory Control) u s V e > V, V e 0. u. u = u c + u s (3.14) (3.14). e = c e + b c [ ( p ( x p ) - p(x)) + ( q( x q ) - q(x )) u c - q(x ) u s ] (3.15) (3.15) (3.12)

28 V e = e T Qe + e T P b c [ ( p ( x p ) - p (x)) (3.16) + ( q( x q ) - q(x )) u c - q(x ) u s ] e T Qe + e T P b c [ p ( x p ) + p (x) + q( x q) u c + q(x )) u c ] - e T P b c q(x) u s, p U, q U, q L (Bounds). p U, q U, q L, (3.16) u s. u s = I * 1 sg n ( e T P b c ) 1 q L ( x ) [ p ( x p ) + p U ( x) + q( x q) u c + q U ( x) u c ] (3.17), V e > V I * 1 = 1, V e V I * 1 = 0. y 0 sg n (y ) = 1, y < 0 sg n (y ) = - 1. V e > V (3.17) (3.16) (3.18) V e 0. V e e T Qe 0 (3.18)

29 3.2.4 u u c u s..,. * p = arg min p p [ s up x U c p( x p ) - p(x) ] (3.19) * q = arg min q q [ s up x U c q( x q) - q(x) ] (3.20), p q p q (Constraint Set)., p q p q. p = { p : p M p } (3.21) q = { q : q M q, y l } (3.22), M p, M q,. (3.1) y l, y l > 0. (Minimum Approximation Error ) = ( p( x * p ) - p( x )) + ( q( x * q ) - q( x)) u c (3.23) (3.15)

30 e = c e - b c q(x) u s + b c [ ( p( x p ) - p ( x p * )) + ( q( x q ) - q( x q * )) u c + ] (3.24) p ( x p ) q( x q ) (3.4) (3.5), (3.24). e = c e - b c q(x) u s + b c + b c [ p T p ( x ) + T q q( x ) u c ] (3.25), p = p - * p, q = q - * q. Lyapunov V = 1 2 e T P e + 1 T p 2 1 p + 1 T q 2 2 q (3.26), 1, 2. (3.25) (3.26) V. V = e T Qe - q(x ) e T P b c u s + e T P b c p T [ p + 1 e T P b c p ( x) ] T q [ q + 2 e T P b c q ( x) u c ] (3.27)

31 , p = p, q = q. (3.27) (3.17) q(x) >0 q(x) e T P b c u s 0.. p = - 1 e T P b c p ( x ) (3.28) q = - 2 e T P b c q ( x ) u c (3.29) (3.27) (3.28) (3.29). V e T Qe + e T P b c (3.30) (3.30) Q 0. e T P b c = 0, p(x ), g (x ) p( x p ), q( x q ), V 0. (Universal Approximation T heorem )

32 p( x p ) q( x q ) IF., Figure Adjustable parameters of a fir st - type indirect adaptiv e fuzzy contr oller

33 U c.., 3.3 U c p ( x p ) q( x q )., U c p ( x p ) q( x q ) Figur e 3.3 Fuzzy rules for a 2nd- order sy stem

34 IF. 3.4 Figur e Adju stable param eter s of a second- type indirect adaptiv e fuzzy contr oller. p, l q F. i. p ( x p ) q( x q )

35 / 1 2/ IF / 1 Figur e 3.5 Adju st able parameter s of a second/fir st - type indirect adaptiv e fuzzy contr oller

36 2/ , 2 IF. 1.., 2/ 1 p, q. p (0) q (0)... 2/

37 ,.. p (0), q( 0). p (0), q (0),.. p, q. p, q. 1, 2. 0,

38 g g m e u c u s : ( ) [rad/ sec] : ( ) [rad] : ( ) [rad] : ( ) [rad] : : u :, u = u c + u s p (x), p (0), q(x ) : q(0) : p, q : : (1 :, 2 : ) 4.1 Figure 4.1 Block diagram of a designed indir ect adaptiv e fuzzy control sy st em for motion contr ol

39 4.2,, 3. s n + k 1 s n k n = 0 k 1,, k n. Q (3.11) Lyapunov P >0. M p, M q,, V. x ( t) u. k, M p, M q,, V. y m, y n m, p U ( x), q U ( x), q L ( x ). F l i. p( x p ) q( x q ). IF i = 1, 2,, n F l i i m 1 m 2 m n. p( x p ) q( x q )

40 R ( l p1,, l pn ) p : IF x 1 is F l p1 1 and and x n is F l pn n THE N p ( x p ) is G ( l p1,, l pn ) (4.1) R ( l q1,, l qn ) q : IF x 1 is F l q1 1 and and x n is F l qn n THE N q ( x q ) is H ( l q1,, l qn ) (4.2) (4.1) (4.2). (4.3). ( l 1,, l n ) ( x) = m 1 l 1 = 1 n i = 1 m n ( l n = 1 F l i (x i i ) n i = 1 F l i(x i i )) (4.3) p ( 0) q (0) p( x p ) q( x q ) (3.4) (3.5). u. u c (3.8) u s (3.17). p ( x p ) q( x q ) (3.4) (3.5).. 1 p q. p q (3.28) (3.29)

41 4.2 Figur e 4.2 Flow chart of a indirect adaptiv e fuzzy contr ol pr ogram

42 DC. e a ( t) = R a i a ( t) + L a i a ( t) + e b ( t) (4.4) T m ( t) = K i i a ( t) (4.5) e b ( t) = K b m ( t) (4.6) T m ( t) = J m m ( t) + B m m ( t) + T L ( t) (4.7) m ( t) = m ( t) (4.8) m ( t) = - K i K b + R a B m R a J m m ( t) + K i R a J m e a ( t) - 1 J m T L ( t) (4.9), R a :, L a :, J m :, B m :, K i :, K b :, e a ( t) :, i a ( t) :, e b ( t) :,

43 m ( t) :, m ( t) :, T m ( t) :, T L ( t) :.. 4.1, (4.8) (4.9). x 1 ( t) = x 2 ( t) (4.10) x 2 ( t) = x 2 ( t) u ( t) - 50 T L ( t) (4.11) y ( t) = x 1 ( t) (4.12), x 1 ( t), x 2 ( t), u ( t), y ( t), T L ( t). 4.1 DC T able 4.1 Parameter s of the DC motor Parameter Value Unit R a 4 ohm L a 0 hen ry J m 0.02 kg m 2 B m 0.01 N m / rad/ sec K i 0.5 N m / A K b 0.5 V/ rad/ sec

44 4.3.2 (4.11) T L ( t). x ( 2) ( t) = p (x) + q(x ) u (4.13) p (x) = x sin (x 1 ) cos (x 2 ) + a 1 ( t) x 1 x 2 (4.14) q(x) = a 2 ( t) (4.15). - 6 x 1 6, x 2 49, 0 u 25 p(x ) x sin (x 1 ) cos (x 2 ) + a 1 ( t) x 1 x q(x ) a 2 ( t) 8.25 q(x ) a 2 ( t) k 1 = 4, k 2 = 4 Q (3.11) P. Q = [ ], P = 36 4 [ 4 5]

45 m 1 = 5, m 2 = 5 m = m 1 m 2 = Figur e 4.3 Member ship function s of a fir st - type fuzzy logic sy stem p (0) [- 5 5], q (0) [5 7.5]

46 2.. (4.16) (4.17) m = 15.,. R ( l m ) p : IF x 1 is F l pm 1 and x 2 is F l pm 2 THE N p ( x p ) is G ( l pm ) (4.16) R ( l m ) q : IF x 1 is F l qm 1 and x 2 is F l qm 2 THE N q ( x q ) is H ( l qm ) (4.17) 15 1, 1. 2/ , 2 1 (4.16) (4.17) /

47 4.4 1 Figur e 4.4 Step respon se of the fir st - type adaptiv e fuzzy contr ol sy stem

48 4.5 1 Figure 4.5 Sinusoidal response of the first- type adaptiv e fuzzy contr ol sy stem

49 4.6 2 Figure 4.6 Step r esponse of the second- type adaptiv e fuzzy contr ol sy stem

50 4.7 2 Figur e 4.7 Sinu soidal respon se of the second- type adaptiv e fuzzy contr ol sy stem

51 4.8 2/ 1 Figure 4.8 Step respon se of the second/fir st - type adaptiv e fuzzy contr ol sy stem

52 4.9 2/ 1 Figur e 4.9 Sinu soidal respon se of the second/fir st - type adaptiv e fuzzy contr ol sy stem

53 5 5.1 Murata ENV - 05DB Gyrostar, DC SPECT ROL 10K., 80C196KC 10bit A/ D., [degree], [degree/ sec].,.,, A/ D.,

54 5.1 Figure 5.1 Output angle and err or of gyro sen sor and potentiomet er

55 5.2 X, Y,., m 1 = 3, m 2 = ( m = m 1 m 2 = 6 ), 5.4 m 1 = 2, m 2 = ( m = m 1 m 2 = 6). p(x ) g (x ) p (0) [- 3 3], q ( 0) [3.5 5] 6 1. A/ D.,, 20[ms] 80C196KC

56 5.2 Figure 5.2 Membership functions of a fuzzy logic system for a roll plant R 1 rolling : IF x 1 is F 1 1 and x 2 is F 1 2 THEN p is G 1 11 and q is H 1 11 R 2 rolling : IF x 1 is F 1 1 and x 2 is F 2 2 THEN p is G 2 12 and q is H 2 12 R 3 rolling : IF x 1 is F 2 1 and x 2 is F 1 2 THEN p is G and q is H R 4 rolling : IF x 1 is F 2 1 and x 2 is F 2 2 THEN p is G 4 22 and q is H 4 22 R 5 rolling : IF x 1 is F 3 1 and x 2 is F 1 2 THEN p is G and q is H R 6 rolling : IF x 1 is F 3 1 and x 2 is F 2 2 THEN p is G 6 32 and q is H Figur e 5.3 Fuzzy rules for a roll plant

57 5.4 Figure 5.4 Membership functions of a fuzzy logic system for a pitch plant R 1 tch in g : IF x 1 is F 1 1 and x 2 is F 1 2 THE N p is G 1 11 and q is H 1 11 R 2 tch in g : IF x 1 is F 1 1 and x 2 is F 2 2 THE N p is G 2 12 and q is H 2 12 R 3 tch in g : IF x 1 is F 1 1 and x 2 is F 3 2 THE N p is G 3 13 and q is H 3 13 R 4 tch in g : IF x 1 is F 2 1 and x 2 is F 1 2 THE N p is G 4 21 and q is H 4 21 R 5 tch in g : IF x 1 is F 2 1 and x 2 is F 2 2 THE N p is G 5 22 and q is H 5 22 R 6 tch in g : IF x 1 is F 2 1 and x 2 is F 3 2 THE N p is G 6 23 and q is H Figur e 5.5 Fuzzy rules for a pitch plant

58 5.6 Figure 5.6 Roll respon se of a plant

59 5.7 Figure 5.7 Roll respon se of a adaptive fuzzy control sy stem

60 5.8 Figure 5.8 Pit ch r espon se of a plant

61 5.9 Figur e 5.9 Pitch respon se of a adaptiv e fuzzy control sy st em

62 [degree]... 15[m s].. 2[degree] ,..., 5.5[degree] 2.5[degree]

63 6,...,.,, 2/ 1.,

64 [1] Charles Broxmeyer, I nertial N avigation Sy s tem s, McGraw - Hill, Inc., [2] C. F. O'Donnell, I nertial N avigation, A naly sis and D es ign, McGraw - Hill, Inc., [3] Kenneth R. Britting, I nertial N avigation Sy s tem s A naly s is, John Wiley & Sons, Inc., [4] B asics of Gy roscop es vol. 1, Applied T echnologies A s sociates, Inc. [5] B asics of Gy roscop es vol. 2, Applied T echnologies A s sociates, Inc. [6],,,, [7] Li- Xin W ang, A dap tive F uz zy Sy stem s and Control : D esig n and S tability A naly s is, Prentice- Hall International, Inc., [8] Li- Xin W ang, "Stable Adaptive Fuzzy Contr ol of Nonlinear Systems," P roc. 31st I E E E Conf. on D ecis ion and Control, T ucson (1992), [9] Intel Corpor ation Liter atur e, E m bedded A pp lications, Intel, [10], 8086 C R OM,, [11],,, [12], FA,,

65 [13],,, [14],, no. 2 : P art,, [15],, no. 3 : P art,, [16] Don Lancaster,, T TL I C,, [17] Intel Corpor ation Literature, 8X C196K C/ 8X C196K D User 's M anual, Intel, [18], M I CR O CON TR OLLER 80196,, [19], 80C196K C, Ohm, [20] Kumpati S. Nar endr a and Anuradha M. Annasw amy, S table A dap tive System s, Prentice- Hall International, Inc., [21] Kevin M. Pas sino and Stephen Yurkovich, F uzzy control, A ddison Wesley Longman, Inc., [22] D. G. Luenberger, L inear and N onlinear P rog ram m ing, A ddison Wesley Publishing Company, Inc., [23] Li- Xin Wang and J. M. Mendel "Fuzzy Basis Functions, Univ er sal Approxim ation, and orthogonal least squares learning," I E E E T rans. on N eural N etwork, vol 3, no. 5 (1992), [24] William L. Br ogan, M od ern Control T heory : T hird edition, Pr entice- Hall International, Inc.,

66 A ppendix A. 80C 196KC M ain Circu it

67 A ppendix B. I/ O Interf ace Circu it